In an example, quantitative image comparison includes a determination of similarities and/or differences among images. Quantitative image comparison may be implemented in various fields, including, e.g., biometrics, automated diagnosis, remote sensing, computer vision, biomedical imaging, and so forth.
In this example, an optimal transportation (OT) metric may be used in quantitative image comparison. Generally, an OT metric includes a value indicative of a distance among images. In this example, the value of the OT metric is inversely correlated to similarity among images. For example, images with decreased OT metrics (e.g., decreased distances) have increased similarity to each other, e.g., relative to similarities among other images with increased OT metrics. Hereinafter, an OT metric may also be referred to as an OT distance, without limitation, and for purposes of convenience.
In an example, the OT metric includes a distance function defined between probability measures on a given set Ω. In this example, if each measure is viewed as a unit amount of mass piled over Ω, the OT metric corresponds to the minimum cost of turning one pile into the other pile. In an example, the cost of transport from one point to another point is the mass transported times the square of the transportation distance. The OT distance squared is the minimum, among all possible ways to turn one pile into the other, of the total cost of transporting all mass. In another example, the piles on M may be represented by images, namely, image μ and image ν. In this example, the OT metric represents the least possible total cost of transporting all of the mass from image μ to image ν.
When the measures to be transported are given as continuous densities then there exists a function φ called the transportation map which transports mass from the first measure to the second measure in an optimal way. Then, if μ has density f then:dOT2(μ,ν)=∫|φ(x)−x|2f(x)dx If the measures μ and ν are discrete particles (as is the case when dealing with images) then one needs to allow for such transformations from μ to ν which can split the particles. Then the optimal transport is given by an optimal transportation plan.
Referring to FIG. 1, representation 10 of an OT plan for images μ, ν is shown. In this example, image μ and image ν both include various locations (e.g., location x) of particles (and/or items of mass). Generally, an OT plan includes a measure onΩ×Ωspecifying for each pair (x,y) how much mass from x is transported toy. In the example of FIG. 1, arrows 12, 14, 16, 18, 20, 22 represent the transportation of particles from image μ to image ν. A system may compute the OT metric between image μ and image ν, e.g., using algorithms that are commonly known in the art.